Research on Dynamical Systems, Chaos and Fractals Modeled by Billiard Systems in Aspect of Global Analysis

改进了二维无月蚀色散台球的Lipschitz连续不变叶化存在性的证明，使其更具建设性和初等性。这可能会在应用上取得新的进展。我们引入了Busemann的测地线几何来研究其位形空间中的台球轨迹。特别地，我们发现了平行线和周期轨迹之间的一些关系。此外，通过计算机模拟，我们观察到台球系统在|x/a|^r+|y/b|^r=1（r>1）的封闭曲线上的有趣现象。它的庞加莱图提供了大量的信息，通过这些信息可以证明这些现象。在被视为经典周期台球玩具模型的双曲和间歇图中，输运被发现是由Frobenius-Perron算子的光谱表征的。进一步研究了边界元法与量子台球散射问题的关系。本文研究了复杂动力学中不同周期点的重整化问题。我们能够定义一个新的映射空间，它在抛物型重整化及其扰动下是不变的。我们观察到可计数到一个分段可逆马尔可夫系统的大偏差。特别地，我们证明了在一定条件下的（二级）大偏差上限和指数递减性质。此外，我们还研究了大偏差规律的多重分形版本。此外，我们还成功地证明了波义耳和马斯的猜想。我们可以构造一个单参数复杂结构族，其临界点在递归点和瞬态切换点之间。我们表明，在这一点上保持重现的变形自由度相对较低。作为与量子混沌等相关的课题，我们一直在研究算子代数本身的理论，并在该课题上取得了一些成果。

We have improved the proof of the existence of Lipschitz continuous invariant foliations for two dimensional dispersing billiards without eclipse and make it more constructive and more elementary. This may give a new progress in applications. We introduced the geometry of geodesies due to Busemann to study the billiard ball trajectories in their configuration spaces. In particular, we found out some relations between parallels and periodic trajectories. Moreover, we observed interesting phenomena of billiard systems in a closed curve given by |x/a|^r+|y/b|^r=1(r>1) with computer simulations. Its Poincare map gives a lot of information, by which those phenomena may be proved.In hyperbolic and intermittent maps regarded as toy models of classical periodic billiards, transports are found to be characterized by the spectra of the Frobenius-Perron operator. Furthermore, we researched the relationship between boundary elements method and scattering problems in quantum billiards.We studied the renormalization associated to indifferent periodic points of complex dynamics. We were able to define a new space of maps which is invariant under parabolic renormalization and its perturbations.We observed large deviations for countable to one piecewise invertible Markov systems. In particular, we showed the(level 2)upper large deviation bounds and exponential decreasing property under certain conditions. Moreover, we researched multifractal version of large deviation laws.Further, we have succeeded to prove the conjecture by Boyle and Maass. We can construct a one-parameter family of complex structures with critical point at the point the recurrence and the transience switch to the other. We showed that freedom of deformation to preserve recurrence at the point is relatively low. As related topics to Quantum Chaos and so on, we have been studying the theory of operator algebras itself and obtained several results in the subject.